What is the distinction between the Byzantine Empire and the Roman Empire?

The Byzantine
Empire is distinguished from the Roman Empire largely by geography, culture, and
religion.

The Byzantine Empire had its origins in the Roman Empire. 
Beginning in the 300s AD, Constantinople essentially became the capital of the Roman Empire in
the East.  This split with Rome became official in 395.  At that point, it was still the Eastern
Roman Empire, not the Byzantine Empire.

The Byzantine Empire came to differ
from the Roman or Western Roman Empire in a number of ways.  Geographically, it was centered in
the East, with the capital in Constantinople which is in what is now Turkey.  The Byzantines did
briefly conquer and control Italy, but for the most part, this was an eastern empire. 
Culturally, the Byzantine Empire was Greek, not Roman, with Latin eventually falling out of use
completely.  In terms of religion, it was built as a Christian empire from its beginnings,
instead of being an empire built on pagan foundations as the West was.

All of
these are ways in which the two empires are distinct even though the one gave rise to the
other.